This paper studies reinforcement learning (RL) when an agent is allowed to look ahead and see which states would result from any short sequence of actions before choosing its next move. The authors show two main things. First, finding the exact optimal plan in this setting is NP-hard even when the usual discount factor (which downweights future rewards) is any fixed rational number between 0 and 1. Second, despite that hardness, they give a randomized polynomial-time approximation scheme (RPTAS) that finds near-optimal policies efficiently when the look-ahead depth ℓ is fixed. They also extend the approach to the learning setting with unknown transitions and noisy rewards and give regret guarantees that match classical results up to logarithmic factors.

Transition look-ahead means the agent is given extra predictive information before acting: at each step it sees the outcomes of every possible sequence of up to ℓ actions starting from the current state. This models real-world situations where limited forward information is available, for example a high-fidelity simulator or real-time traffic predictions that show the effect of a short sequence of routing choices. Technically, this extra information enlarges the state description and can change the planning problem in important ways.

On the negative side, the paper strengthens earlier hardness results. Previous work had shown NP-hardness when the discount factor was allowed to be arbitrarily close to 1. Here the authors prove that exact planning remains NP-hard for every fixed rational discount factor γ in (0,1). In plain terms, computing the truly optimal policy that fully uses multi-step look-ahead is intractable in the worst case, under standard complexity assumptions.

On the positive side, the authors provide an algorithmic fix for practical purposes. For any fixed look-ahead depth ℓ, they design a randomized polynomial-time approximation scheme (RPTAS) that, with high probability, builds a policy whose performance is uniformly close to optimal across all possible look-ahead windows. Their approach replaces the Bellman expectations by a fixed sample of complete transition tables and solves the resulting empirical planning problem once as preprocessing. For fixed ℓ this gives polynomial-time computation in the problem size.